32 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA
(a) If the function / is differentiate at a point x e U, then D+/(*) = || /'(*)!!• If
f',(x) is not a linear homeomorphism of E onto a subspace of F, we have D~/(*) — 0-
In the contrary case, we have D~/W = II (/'(*))" I"1, where (Ax))'1 denotes the
(b) Suppose that the segment with endpoints a, b is contained in U, and that
D+/(*) ^ M for all points x in this segment. Prove that \\f(b) -f(a)\\ ^ M\\b - a\\
(see the proof of 8.5.1).)
(c) Take U = E =R2 and F =R3, and let /be defined as follows :
if (&» £2) ^ (0, 0); and/(0, 0) = 0. Show that the greatest lower bound of D~/on U
is strictly positive, but that there is no neighborhood of 0 in U on which /is injective.
(d) For the remainder of this problem, suppose that there exist two numbers m, M
with 0<m<M<+oo, such that m ^ D~/W 2 M for all x e U. Suppose also that
for each jceU there exists an open neighborhood V of *4n U such that/|V is a
homeomorphism of V onto an open set in F; then/(U) is open in F. Let a be a point
of U, and for each line D c F containing f(a) let ID be the connected component of
the point f(a) in the open set D n /(U) in D. The union Sa of the sets ID is the largest
star-shaped open set with respect to /(a) contained in /(U). For each line D in F
passing through f(a), there exists a unique continuous mapping &D : ID -> U such that
#D(/(#)) = a and f(g^(y)) = y for all y e ID (the proof is the same as in Section 1 0.2,
Problem 6(c).) If y9y' are points of ID, we have || #>(/)— #>(>) II ^s^"1!!/ — y\\-
Deduce that as y tends to an endpoint of ID (when there is one), gD(y) has a limit
belonging to Fr(U).
(e) Let y : J -> U be a path in U with origin a and extremity b. Show that, if
/(ytO) c Sfl, then we have b =#D(/(W), where D is the line through f(a) and/(6), and
(f) Deduce from (d) and (e) that if U = E then Sfl = F, and /is a homeomorphism
of E onto F.
(g) Let k= M/77J. Let a, b be two points of U such that the set Ekta>b of points
z e E such that \\z - a\\ + \\z-b\\£ k\\ b - a\\ is contained in U. If L is the closed
segment with endpoints a, b, show that/(L) c: Sfl , and hence that
(Proof by contradiction: consider the least / e [0, 1] such that y =f(a + t(b — a)) $ Sa .
If D is the line through f(a) and y, there is an endpoint u of ID belonging to the open
segment with endpoints f(a) and y. When t' < t tends to /, there is a point u' of the
open segment with endpoints f(a) and y' = f(a + t'(b — a)) which tends to u. Let D'
be the line through f(a) and /, and let z' = gw(u'). Using (e), show that z' e Eki a, b ,
and obtain a contradiction by making t' tend to t and using (d).)
(h) Suppose that E and F are Hilbert spaces and that U is the ball ||x|| < 1. Deduce
from (g) that if B is the ball ||*|| < 1/(1 + (k2 - I)1/2), the restriction of /to B is a
homeomorphism onto/(B), and that ||/(^) -/(x)|| ^ m \\x' ~ x\\ for any two points
x, x' in B.
(i) With the hypotheses of (h), suppose also that A: < ((1 +-\/5)/2)1/2. Show that /
is then infective on U and, more precisely, that for any two points x, x' in U we have
) -/(x)j| ^ /x ||^ - x|| , where
_ m2 ~ M(M2 - w2)1/2nd (3.13.4)). To get corresponding results in general, it is